Kramers–Kronig relations

Summary

The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often used to compute the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the condition of analycity, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics, these relations are known by the names Sokhotski–Plemelj theorem and Hilbert transform. Formulation Let \chi(\omega) = \chi_1(\omega) + i \chi_2(\omega) be a complex function of the complex variable \omega, where \chi_1(\omega) and \chi_2(\omega) are real. Suppose this function is analytic in the closed upper half-plane of \omega and diminishes faster than

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https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations

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A conformal dispersion relation: correlations from absorption

Din Carmi

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its "absorptive part", defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the "inverted" conformal block with the ordinary conformal block.

2020

, ,

Undoped fluoride glass slides have been exposed to pulsed 193-nm ultraviolet (UV) irradiation. Their absorption changes have been measured to evaluate UV-induced index changes using Kramers-Kronig relation. A layer-peeling polishing technique was applied to characterize the local UV-induced index change of highly absorbing glass. Index changes up to 1.75 x 10(-4) have been evaluated with this method in fluorozirco-aluminate glass. Fluoroaluminate and fluorozirconate glass showed only small index changes of about 2.0 x 10(-6) and 2.6 x 10(-6) at a wavelength of 1550 nm.

2005

Wave dispersion in periodic post-buckled structures

Florian Paul Robert Maurin, Alessandro Spadoni

Wave propagation in pinned-supported, post-buckled beams can be described with the Korteweg de Vries (KdV) equation. Finite-element simulations however show that the KdV is applicable only to post-buckled beams with strong pre-compression. For weak and moderate pre-stress, a dispersive front is present and it is the aim of the current paper to analyze sources of dispersion beyond periodicity given three support types: guided, pinned, and free. Bloch theorem and a transfer-matrix method are employed to obtain numerical dispersion relations and characteristic wave modes, which are used to analyze the effects of pre-stress, initial curvature, and the influence of support types. Additionally, a new method is proposed to obtain a semi-analytical dispersion equation for the acoustic branch. Powers of frequency and the propagation constant are explicitly expressed and their coefficients are based on stiffness and mass-matrix components obtained from finite elements. This allows a physical interpretation of the dispersion sources, based on which, equivalent mass-spring models of post-buckled beam are proposed. It is found that mass and stiffness coupling are significant dispersion sources. In the present paper, a reduced form of Bloch theorem is presented exploiting glide-reflection symmetries, reducing the size of the unit cell and allowing an easier representation and interpretation of results.

Elsevier2014